Conservative Forces and Potential Energy

Conservative Forces and Potential Energy

A force can transfer energy to or from a particle by doing work. For a special and very important class of forces, the work between two positions depends only on the endpoints and not on the path followed between them. Such a force is called conservative.

For a conservative force, the work done from point A to point B can be represented by a scalar function of position called the potential energy U:

(1)

Equivalently,

(2)

This relation is the central idea of the article. It turns certain force calculations into energy bookkeeping and prepares the way for conservation of mechanical energy.

1 What makes a force conservative?

A force is conservative if the work it does between two fixed positions is independent of the path.

If a particle moves from point A to point B along two different paths C₁ and C₂, then a conservative force satisfies

(3)

Therefore the work can depend only on the endpoints A and B.

This is a strong restriction. Many familiar forces are conservative in idealized mechanics, including

• uniform gravity near Earth’s surface,
• Newtonian inverse-square gravity,
• the force of an ideal spring,
• the electrostatic Coulomb force.

Common examples of nonconservative forces include kinetic friction and many forms of drag.

2 Closed-path test

Suppose a conservative force carries a particle from A to B along one path and the particle returns from B to A along another path.

Because the work between fixed endpoints is path independent,

(4)

Therefore the total work around the closed path is zero:

(5)

For the class of force fields considered in elementary mechanics, the following statements are equivalent:

1. the work between two points is path independent,
2. the work around every closed path is zero,
3. a potential-energy function can be defined,
4. the force can be obtained from the spatial derivative of that potential energy.

The last statement will be developed carefully below.

3 Defining potential-energy difference

Let a conservative force do work

(6)

Define the change in potential energy by

(7)

Thus

(8)

The minus sign is essential.

If the conservative force does positive work,

(9)

then

(10)

The potential energy decreases.

If the conservative force does negative work,

(11)

then

(12)

The potential energy increases.

4 Potential energy is defined up to a constant

Only differences in potential energy are determined by force.

If U(r) is a valid potential-energy function, then

(13)

is equally valid for any constant C.

This is because

U′B − U′A	= (UB + C) − (UA + C)	(14)
	= UB − UA.	(15)

Therefore the physics is unchanged.

One is free to choose a convenient reference level at which

(16)

For uniform gravity near Earth’s surface, the reference height can be chosen arbitrarily. For Newtonian gravity, a common and useful convention is

(17)

5 One-dimensional force from potential energy

In one-dimensional motion,

(18)

Divide by dx:

(19)

This equation contains a powerful geometric idea.

The force points in the direction in which potential energy decreases most rapidly.

If

(20)

then

(21)

If

(22)

then

(23)

If

(24)

then

(25)

6 Recovering potential energy from force

If the force is known, potential energy can be found by integration.

From

(26)

integrate from a reference point x₀ to x:

(27)

Therefore

(28)

The variable x′ is a dummy integration variable. It prevents confusion between the integration variable and the endpoint x.

7 Example 1: constant force

Suppose

(29)

is constant.

Then

U(x) − U(x0)	= −∫ x0xF 0 dx′	(30)
	= −F0(x − x0).	(31)

Thus

(32)

where

(33)

A constant force therefore corresponds to a linear potential-energy function.

8 Uniform gravity near Earth’s surface

Near Earth’s surface, take the y axis positive upward.

The gravitational force on a mass m is

(34)

In one dimension,

(35)

The potential-energy derivative satisfies

(36)

Therefore

(37)

so

(38)

Integrating,

(39)

If the zero of potential energy is chosen at y = 0, then

(40)

The difference between two heights is

(41)

The work done by gravity is

(42)

9 Example 2: lifting a mass

A 3.0 kg mass is raised vertically from y_i = 1.0 m to y_f = 6.0 m.

The change in gravitational potential energy is

ΔUg	= mg(yf − yi)	(43)
	= (3.0)(9.81)(5.0)	(44)
	= 147 J.	(45)

Therefore gravity does

(46)

The result depends only on the change in height. It does not depend on whether the mass moved vertically, along a ramp, or along some other path, provided gravity is the same uniform conservative field.

10 Elastic potential energy of an ideal spring

For an ideal spring,

(47)

Use

(48)

Then

(49)

so

(50)

Integrating,

(51)

Choosing

(52)

gives

(53)

This is the elastic potential energy stored by an ideal spring.

11 Work and spring potential energy

The spring work from x_i to x_f is

(54)

Therefore

Ws	= −	(55)
	= kxi2 −kxf2 .	(56)

This reproduces the result derived directly from the work integral in M03-03.

The energy viewpoint and the force-integral viewpoint are not separate physical laws. They are two descriptions of the same conservative interaction.

12 Newtonian gravitational potential energy

Consider two masses M and m separated by distance r.

The gravitational force on m is radial and inward:

(57)

For radial motion,

(58)

The potential satisfies

(59)

Thus

(60)

so

(61)

Integrating,

(62)

With the standard reference

(63)

we obtain

(64)

The negative sign reflects the chosen zero at infinite separation. At any finite separation, the gravitational potential energy is lower than at infinity.

13 Why near-Earth gravity gives mgy

The inverse-square potential is

(65)

Near Earth’s surface, write

(66)

where

(67)

A first-order expansion gives

(68)

Therefore

U(RE + y)

≈− + y.

(69)

Since

(70)

this becomes

(71)

The first term is a constant reference offset, so locally

(72)

Thus the familiar near-Earth formula is the local approximation to the inverse-square gravitational potential.

14 Potential energy in several dimensions

In three-dimensional Cartesian coordinates,

(73)

For a conservative force,

(74)

Since

(75)

we have

(76)

But the total differential of U(x,y,z) is

(77)

Comparing coefficients,

Fx	= −,	(78)
Fy	= −,	(79)
Fz	= −.	(80)

Therefore

(81)

The gradient is

(82)

The force points toward decreasing potential energy.

15 Equipotential surfaces

A surface on which

(83)

is called an equipotential surface.

For a displacement tangent to an equipotential surface,

(84)

Since

(85)

we obtain

(86)

Thus the conservative force is perpendicular to an equipotential surface.

For Newtonian gravity around a spherical mass, the equipotential surfaces are spheres centered on the mass.

16 Potential-energy curves

In one-dimensional problems, a graph of U(x) reveals important dynamical information.

Because

(87)

the slope tells us the force direction.

A local minimum of U satisfies

(88)

and typically corresponds to stable equilibrium.

A local maximum also satisfies

(89)

but it typically corresponds to unstable equilibrium.

17 Stable equilibrium

Suppose x = x₀ is a local minimum of the potential energy.

Then

(90)

so

(91)

For a small displacement to the right of x₀, the potential rises:

(92)

Therefore

(93)

which points back toward equilibrium.

For a small displacement to the left,

(94)

so

(95)

again pointing back toward equilibrium.

This is why a potential-energy minimum is stable.

A mathematical test is

(96)

18 Unstable equilibrium

At a local maximum,

(97)

but

(98)

A small displacement produces a force that pushes the particle farther away from the equilibrium point.

Thus a local maximum corresponds to unstable equilibrium.

19 Example 3: equilibrium from a potential

Let

(99)

where

(100)

Equilibrium occurs where

(101)

Differentiate:

(102)

Factor:

(103)

Thus the equilibria are

(104)

and

(105)

The second derivative is

(106)

At

(107)

we have

(108)

so the origin is unstable.

At

(109)

we obtain

U′′	= 12a− 2b	(110)
	= 4b > 0.	(111)

These two equilibria are stable.

20 Conservative and nonconservative forces

A conservative force stores and releases mechanical energy through a potential-energy function.

A nonconservative force generally cannot be represented globally by a single-valued position-only potential energy.

For example, kinetic friction usually does

(112)

over a path of nonzero length.

If the particle returns to its starting position after traveling around a loop, friction has generally done net negative work:

(113)

Therefore friction fails the closed-path test for a conservative force.

The energy removed from mechanical motion is not destroyed. It is transferred into thermal energy, deformation, sound, and microscopic degrees of freedom.

21 Path independence versus path length

A common source of confusion is to think that conservative work depends on path length.

It does not.

Consider uniform gravity. A particle may descend from height y_i to height y_f by

• falling vertically,
• sliding along a straight incline,
• moving along a curved track.

If only the gravitational work is being calculated, all three paths give

(114)

The actual distances traveled can be different, but the gravitational work depends only on the endpoint heights.

22 Dimensional checks

Potential energy has the same dimensions as work:

(115)

For uniform gravity,

(116)

so

(117)

For a spring,

(118)

Since

(119)

we obtain

(120)

For inverse-square gravity,

(121)

which also has units of joules.

23 Common mistakes

1. Forgetting the minus sign in ΔU = −W_c.
2. Assuming every force has a potential-energy function.
3. Treating the absolute value of potential energy as physically unique.
4. Forgetting that only potential-energy differences affect mechanics.
5. Writing F = dU∕dx instead of F = −dU∕dx.
6. Confusing stable equilibrium with any point where U′(x) = 0.
7. Assuming a local maximum of U is stable.
8. Using mgy over altitude ranges where constant g is no longer a good approximation.
9. Forgetting that U = −GMm∕r uses the reference U(∞) = 0.
10. Assuming path-independent gravitational work means the particle travels the same distance along every path.
11. Treating frictional work as if it could be represented by a position-only potential energy.

24 Practice exercises

1. A conservative force does +25 J of work on a particle. Find the change in potential energy.
2. A conservative force does −18 J of work. Does the potential energy increase or decrease, and by how much?
3. A force has potential energy U(x) = 4x² in SI units. Find F _x(x).
4. A force is F_x = −6x². Find a corresponding potential-energy function U(x).
5. A 2.0 kg mass moves upward by 3.0 m near Earth’s surface. Find ΔU_g.
6. A spring with k = 250 N∕m is compressed by 0.12 m. Find its elastic potential energy relative to equilibrium.
7. Starting from F_x = −kx, derive U_s = kx² + C.
8. Starting from F_r = −GMm∕r², derive U _g = −GMm∕r + C.
9. Show that adding a constant C to a potential-energy function does not change the force.
10. For U(x) = ax⁴ − bx² with a,b > 0, classify all equilibrium points.
11. Explain why the gravitational work between two fixed heights is the same on a straight ramp and a curved track.
12. A particle moves around a closed loop under a force field. The measured net work is 12 J. Can the force be conservative? Explain.
13. A potential-energy curve has negative slope at some point. What is the sign of the force there?
14. For U(x) = U₀ cos(kx), derive F_x(x) and identify equilibrium positions.
15. Explain physically why a local minimum of potential energy corresponds to stable equilibrium.

25 Summary

A conservative force does path-independent work:

(122)

Equivalently,

(123)

For a conservative force around a closed path,

(124)

In one dimension,

(125)

In three dimensions,

(126)

Important examples are

(127)

near Earth’s surface,

(128)

for an ideal spring, and

(129)

for Newtonian gravity with U(∞) = 0.

Potential-energy curves also reveal force direction and equilibrium:

(130)

at equilibrium, with

(131)

for stable equilibrium.

The next article, M03-06, combines kinetic and potential energy into the conservation of mechanical energy.

References

References